Let N=22*31 + 11*17 + 13*19. Without actually computing N, determine the parity of N, the units digits of N, and the remainder when N is divided by 7.

parity of N is even.. since, odd*odd=odd, even*odd=even, odd+odd=even and odd+even=odd..
the unit digit is 6.. since 2*1 + 1*7 + 3*9, and 3*9=27, hence, you just need to add 2+7+7=16.. Ü
for the remainder, i had a long method. you just need to express N=7p+r and by Division Algo., r is the remainder which is in that case, r=3. Ü